Car trip without living in the car

The better way might be to do four nights in Dublin and a day trip by train to Belfast or drive it if you want. That would leave you with six more nights and I might suggest one in two in Galway (Cliffs of Moher, Aran Islands, The Burren), two in Killarney (Galway-Killarney is a long stretch) for Ring of Kerry and Dingle Penninsula and one in Cork followed by one in Dublin before your departure. An alternative is to lose a Dublin night, add it to Belfast but the drive from Belfast to Galway is five hours via greater Dublin.

All in all, you are pushing it maybe for driving times but not to breaking point in my opinion.

While in Dublin, you shouldn't use a car. Dublin is quite compact, walking from the major sights north to south or east to west will be less than 30 minutes at most and there is a good public bus service. Pick up a Leap card at the airport and use it like an Oyster/|Octopus/Myki card.

If you have the option you could fly into Dublin and out of Cork depending on where you are going after Ireland which would give you and extra night somewhere.

Train or coach for a day trip to Belfast. Belfast city centre is compact and thee are good bus services to the suburbs. You can then limit car hire just to the southern loop of your proposed trip.

If wanting to reduce the amount of driving you could also use train or coach to say Limerick or Cork and hire a car from there. Relaxing train journey on modern trains.

R

I would say no need to hire a car for the Dublin part of your trip.
If your 3 nights in Dublin are at the beginning or end of the trip which they probably are then only rent the car for the time outside Dublin. Cork is only 2.5 hours from Dublin on a good highway as long as you don't travel in the main commuter rush times of day.
You could certainly do Belfast from Dublin by train. Make your Dublin stay a 4 day and make one of them a daytrip to Belfast?